\begin{section}{Complex Numbers}

\begin{quotation}
``{\it Where the Mystery is the deepest is the gate of all that is subtle and wonderful.}''
\end{quotation}

The Complex Plane is a strange place where certain properties that we know about Real Numbers appear to hold, but there also this exotic ``imaginary axis'' that intersects the real line which introduces (quite literally) a new dimension for some very interesting things to happen that we do not see elsewhere.\\

Complex Analysis is the study of functions on the complex plane, fundamental discoveries in this field have been made throughout the last few centuries ago great mathematicians such as Tartaglia, Ferro, Cardano, Bombelli, Euler, Riemann, Cauchy, and Gauss.\\

A complex number $z \in \mathbb{C} $ is typically defined as any number of the form $\alpha + i\beta$ where $i = \sqrt{-1}$ is the imaginary unit and $\alpha, \beta$ are real numbers. A more formal definition of complex numbers will be given in this section. \\

As a brief historical note, the complex numbers were first investigated by an Italain mathematician called Girolamo Cardano in 1545 as a necessary ingredient to his method of solving {\bf cubic equations} of the form:

\begin{equation}
z^3 + a_{2}z^2 + a_{1}z + a_{0} = 0.
\end{equation}

Cardano discovered that the substitution $z = x - \frac{a_{2}}{3}$ to transforms the equation into a {\bf depressed cubic}

\begin{equation}
x^3 + bx + c = 0.
\end{equation}

\hspace{2mm}

where $b = a_{1} - \frac{1}{3}a{2}^{2}$ and $c = -\frac{1}{3}a_{1}a_{2} + \frac{2}{27}a_{2}^{3} + a_{0}.$ \\

Then any solution for $x$ gives a solution for the general cubic equation. \\

The method of solving depressed cubics was well-known to Cardano, and was developed three decades earlier by the mathematicians Ferro and Tartaglia. Incidentally, general solution to the depressed cubic is given by the {\bf Ferro-Tarataglia Equation}:

\begin{equation}
x = \sqrt[3]{-\frac{c}{2} + \sqrt{\frac{c^2}{4} + \frac{b^3}{27}}} + \sqrt[3]{-\frac{c}{2} - \sqrt{\frac{c^2}{4} + \frac{b^3}{27}}}.
\end{equation}

We now move on to some basic properties of the complex numbers which lay the foundation for further study.

\begin{subsection}{The Algebra and Geometry of Complex Numbers}

We can think of a complex number $z$ as an ordered pair of real numbers $(x,y)$ where $z = x+iy$. This forms a conceptual basis for the coordinate system used for the complex plane, and the algebra of complex numbers can be thought of as running parallel to the algebra of vectors. \\

So, how are complex numbers added and subtracted? \\

Let $z_{1},z_{2} \in C$ be complex numbers such that $z_1 = x_1 + iy_1$ and $z_2 = x_2 + iy_2$. Then their sum is easily computed

\begin{equation}
z_1 + z_2 = (x_1,y_1) + (x_2,y_2) = (x_1+x_2,y_1+y_2).
\end{equation}

Similarly, the difference of two complex numbers is

\begin{equation}
z_1 - z_2 = (x_1,y_1) - (x_2,y_2) = (x_1-x_2,y_1-y_2).
\end{equation}
 
\begin{ex}
Compute the sum and difference of $z_1 = 3+2i$ and $z_2 = 4+3i$.
$$z_1 + z_2 = (3,2) + (4,3) = (3+4,2+3) = (7,5) = 7 + 5i.$$
$$z_1 - z_2 = (3,2) - (4,3) = (3-4,2-3) = (-1,-1) = -1 - i.$$
\end{ex}

Addition and substract of complex numbers is easy enough then. How about multiplication and division? \\ 

It turns out that this is a bit more tricky, since the requirement $i^{2} = -1$ forces us into a somewhat counter-intuitive definition of complex multiplication. However, it is not difficult to convince oneself that the product of two complex numbers can be computed as

\begin{equation}
z_{1}z_{2} = (x_1,y_1)(x_2,y_2) = (x_{1}x_{2} - y_{1}y_{2}, x_{1}y_{2} + x_{2}y_{1}).
\end{equation}

Notice the similarity to the vector cross product in $\mathbb{R}^2$.

\begin{ex}
Compute the product of $z_1 = 3+2i$ and $z_2 = 4+3i$.
$$z_{1}z_{2} = (3,2)(4,3) = ((3)(4) - (2)(3),(3)(3) + (2)(4)) = (12 - 6, 9 + 8) = (6,17) = 6 + 17i.$$
\end{ex}

Division of complex numbers is equally strange. Assume that $z_2 \neq 0$, then

\begin{equation}
\frac{z_1}{z_2} = \frac{(x_1,y_1)}{(x_2,y_2)} = (\frac{x_{1}x_{2} + y_{1}y_{2}}{x_{2}^{2} + y_{2}^{2}}, \frac{-x_{1}y_{2} + x{2}y_{1}}{x_{2}^{2} + y_{2}^{2}}.
\end{equation}

\begin{ex}
Compute the quotient of $z_1 = 3 + 7i$ by $z_2 = 5 - 6i$.
$$\frac{z_1}{z_2} = \frac{(3,7)}{(5,-6)} = (\frac{15 - 42}{25 + 36}, \frac{18+35}{25+36}) = (\frac{-27}{61},\frac{53}{61}).$$
\end{ex}

Algebraically speaking, the complex plane is a {\bf field}. This means it is a set endowed with two binary operations (addition and multiplication) which obey the following basic axioms:

\begin{enumerate}
\item {\bf Commutative Law for Addition:} $z_1 + z_2 = z_2 + z_1.$
\item {\bf Associative Law for Addition:} $z_1 + (z_2 + z_3) = (z_1 + z_2) + z_3.$
\item {\bf Additive Identity:} There exists $\omega \in \mathbb{C}$ such that $z + w = w + z = z$ for all $z \in \mathbb{C}$.
\item {\bf Additive Inverses:} For any $z \in \mathbb{C}$, there exists a unique $(-z) \in \mathbb{C}$ such that $ z + (-z) = (-z) + z = \omega$.
\item {\bf Commutative Law for Multiplication:} $z_{1}z_{2} = z_{2}z_{1}.$
\item {\bf Associative Law for Multiplication:} $z_{1}(z_{2}z_{3}) = (z_{1}z_{2})z_{3}.$
\item {\bf Multiplicative Identity:} There exists $\zeta \in \mathbb{C}$ such that $z \zeta = \zeta z = z$ for all $z \in \mathbb{C}$.
\item {\bf Multiplicative Inverses:} For any $z \in \mathbb{C}$, there exists a unique $z^{-1}$ such that $zz^{-1} = z^{-1}z = \zeta$.
\item {\bf Distributive Law:} $z_1(z_2 + z+3) = z_{1}z_{2} + z_{1}z_{3}.$
\end{enumerate}

Note that the additive identity is obviously $\omega = (0,0) = 0$ and the multiplicative identity is $\zeta = (1,0) = 1$. \\

These are simple extensions of the properties of the field of Real Numbers to the Complex Numbers and are not particularly interesting. The three definitions that follow, however, are unique to complex numbers and allow us to postulate an interesting theorem.

\begin{defn}
The {\bf real part} of a complex number $z = x + iy$, denoted as $\Re(z)$, is the real number $x$.
\end{defn}

\begin{defn}
The {\bf imaginary part} of a complex number $z = x + iy$, denoted as $\Im(z)$, is the real number $y$.
\end{defn}

\begin{defn}
The {\bf conjugate} of a complex number $z = x + iy$, denoted as $\bar{z}$, is the complex number $\bar{z} = x - iy$.
\end{defn}

\begin{ex}
Let $z = -3 + 9i$. Then $\Re(z) = -3$, $\Im(z) = 9$, and $\bar{z} = -3 - 9i$.
\end{ex}

Now we state a theorem about these fundamental definitions.

\begin{thm}
Suppose $z,z_1,z_2 \in \mathbb{C}$ are complex numbers. Then
\begin{itemize}
\item $\displaystyle \overline{\overline{z}} = z.$
\item $\displaystyle \overline{z_1 + z_2} = \bar{z_1} + \bar{z_2}.$
\item $\displaystyle \overline{z_1 z_2} = \bar{z_1} \bar{z_2}.$
\item $\displaystyle \overline{\frac{z_1}{z_2}} = \frac{\bar{z_1}}{\bar{z_2}}.$
\item $\displaystyle \Re(z) = \frac{z + \bar{z}}{2}.$
\item $\displaystyle \Im(z) = \frac{z - \bar{z}}{2i}.$
\item $\displaystyle \Re(iz) = -\Im(z).$
\item $\displaystyle \Im(iz) = \Re(z).$
\end{itemize}
\end{thm}

These relations are relatively obvious and will not be proved here. Instead, let us introduce one more concept and state some theorems about the geometry of complex numbers.

\begin{defn}
The {\bf modulus} of a complex number $z = x + iy$, denoted as $|z|$, is the non-negative real number defined by $|z| = \sqrt{x^2 + y^2}.$
\end{defn}

The modulus represents the distance of the complex number $z$ from the origin $(0,0)$, and the numbers $|z|, |\Re(z)|, |\Im(z)|$ are all lengths of the sides of a right triangle where $|z|$ is the length of the hypotense. \\

From this, it can be deduced algebraically that

\begin{equation}
|x| = |\Re(z)| \leq |z|,
\end{equation}

\begin{equation}
|y| = |\Im(z)| \leq |z|.
\end{equation}

The difference $z_2 - z_1$ represents the displacement vector between the points $z_1$ and $z_2$. The magnitude of the displacement, or the {\bf distance} between the two points is given by the familiar relation:

\begin{equation}
|z_1 - z_2| = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2)}.
\end{equation}

We see also that if $z$ defines a point in the plane, then $-z$ represents the reflection of $z$ through the origin and $\bar{z}$ represents the reflection of $z$ about the real axis. \\

The next equation is trivial to prove but also very important, and shows how the modulus and conjugate of a complex number are related.

\begin{equation}
|z|^2 = z \bar{z}
\end{equation}

This brings us to the statement of another useful and important theorem (which again, will not be proved here):

\begin{thm}{The Triangle Inequality}
Suppose $z_1$ and $z_2$ are arbitrary complex numbers. Then
\begin{equation}
|z_1 + z_2| \leq |z_1| + |z_2|
\end{equation}
\end{thm}

This inequality yields several useful identities.

\begin{cor}
\begin{itemize}
\item $|z_1| = |(z_1 + z_2) + (-z_2)| \leq |z_1 + z_2| + |-z_2| = |z_1 + z_2| + |z_2|.$
\item $|z_1 + z_2| \geq |z_1| - |z_2|.$
\item $|z_1 z_2|^2 = (z_1 z_2) \overline{(z_1 z_2)} = |z_1|^2 |z_2|^2.$
\item $|z_1 z_2| = |z_1| |z_2|.$
\end{itemize}
\end{cor}

Although we have been using the notation $z = x + iy$ to express complex numbers, this is not the only way they may be written. The following definitions will formalize our notation for complex numbers and demonstrate the geometry that is ``built in'' to the complex numbers.

\begin{defn}
A complex number can be written in {\bf Cartesian Form} (or {\it Rectangular Form}) as $$z = x + iy.$$
\end{defn}

What this means geometrically is we can plot a complex number by taking $\Re(z)$ as the $x$-coordinate on the complex plane and $\Im(z)$ as the $y$-coordinate and drawing a line to this point from the origin. Note that this is exactly how vectors in $\mathbb{R}^2$ are drawn. \\

This is nothing new or surprising, but as we will see the Cartesian notation does not tell us the full story about a complex number. Now we introduce an alternative notation for complex numbers.

\begin{defn}
A complex number can be written in {\bf Polar Form} as $$z = re^{i\theta}.$$
Where $r = |z|$ and $\theta$ is the angle between $z$ and the positive real axis.
\end{defn}

This deserves some geometric explanation. In one notation, we think of a complex number as a line from the origin to a point given by the rectangular coordinates $(x,y)$ in a manner roughly equivalent to vectors in Euclidean Space. In the other notation, we think of a complex number as a line with a given length (or radius) $r$ at a certain angle with the positive real line. 

\begin{prop}
Using the general equation to transform Cartesian Coordinates into Polar Coordinates, we can write
$$z = x + iy = (r\cos\theta, r\sin\theta) = r(\cos\theta + i\sin\theta).$$
\end{prop}

The following equation is an immediate consequence of this proposition. \\

Let $z \in \mathbb{C}$ such that $z = re^{i\theta} = r(\cos\theta + i\sin\theta).$ Then

\begin{equation}
e^{i\theta} = \cos\theta + i\sin\theta.
\end{equation}

This remarkable equation is known as {\bf Euler's Formula} and is used extensively in complex analysis and beyond. 

Using this we can easily derive {\bf Euler's Identity} $$e^{i\pi} + 1 = 0.$$

There is now a sudden interest in the set of possible values of $\theta$, which motivates the next definition.

\begin{defn}
Let $z \in \mathbb{C}$ and $z \neq 0$. Define the set
\begin{equation}
{\rm arg}(z) = { \theta \colon z = r(\cos\theta + i\sin\theta)}.
\end{equation}
Then we call any $\theta \in {\rm arg}(z)$ {\bf an argument} of $z$.
\end{defn}

Canonically, mathematicians like to narrow down values of $\theta \in {\rm arg}(z)$ to the interval $[-pi, \pi]$.

\begin{defn}
Let $z \in \mathbb{C}$ and $z \neq 0$. Then $${\rm Arg}(z) = \theta,$$ provided $z = r(\cos\theta + i\sin\theta)$ and $-\pi < \theta \leq \pi.$ \\
If $\theta = {\rm Arg}(z)$ then $\theta$ is {\bf the argument} of $z$.
\end{defn}

\begin{ex}
$${\rm Arg}(1 + i) = \frac{\pi}{4}.$$
\end{ex}

\begin{rmk}
Clearly, if $z = x + iy = r(\cos\theta + i\sin\theta$, and $x \neq 0$, then
$${\rm arg}(z) \subset \arctan \frac{y}{x},$$ where $\arctan \frac{y}{x} = { \theta \colon \tan\theta = \frac{y}{x}}.$
\end{rmk}

The function ${\rm Arg}(z)$ is discontinuous since it ``jumps'' by $2\pi$ each time $z$ crosses the the negative real line. 

We are now prepared to do some more sophisticated algebra with complex numbers. A key ingredient in finding roots of equations with complex coefficients is the fundamental theorem of algebra.

The following theorem is in fact a corollary to the fundamental theorem of algebra.

\begin{thm}
If $P(z)$ is a polynomial of degree $n > 0$ with complex coefficients, then the equation $P(z) = 0$ has precisely $n$ (not necessarily distinct) solutions.
\end{thm}

Consider the roots of the equation $z^n = 1$ where $n$ is an integer. If we can find $n$ distinct solutions, then we have found all possible solutions. However, it is not always easy to tell what constitutes a ``distinct'' solution.

To resolve this dilemma, we say $z_1 = z_2$ if and only if $r_1 = r_2$ and $\theta_{1} = \theta_{2} + 2 \pi k$, where $k$ is an integer.

As it turns out, we can express all $n$ solutions to $z^n = 1$ as

\begin{equation}
z_k = e^{i\frac{2k\pi}{n}} = \cos(\frac{2k\pi}{n}) + i\sin(\frac{2k\pi}{n}),
\end{equation}

for $k = 0,1,2,\dots,n-1.$

These solutions are called the {\bf $n$th roots of unity}.

As we know from abstract algebra, the $n$th roots of unity forms a cyclic group of order $n$ under multiplication. This group is denoted as $U_n$ and is generated by the element $$\omega_n = e^{i\frac{2\pi}{n}} = \cos\frac{2\pi}{n} + i\sin\frac{2\pi}{n}.$$

This element $\omega_n$ is called the {\bf primitive $n$th root of unity}.

\end{subsection}

\end{section}
